Question

For a first-order reaction, how long will it take for the concentration of reactant to fall to one-eighth its original value? Express your answer in terms of the half-life \(\left(t_{1 / 2}\right)\) and in terms of the rate constant \(k\).

Short answer

The time is \( 3t_{1/2} \) or \( \frac{2.079}{k} \) for the concentration to fall to one-eighth.

Step-by-step solution

Understanding First-Order Reaction

In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant. The rate equation for a first-order reaction can be expressed as \( [A] = [A_0] e^{-kt} \), where \( [A] \) is the concentration at time \( t \), \( [A_0] \) is the initial concentration, \( k \) is the rate constant, and \( t \) is the time.

Relating Concentration to Half-Life

The half-life \( t_{1/2} \) of a first-order reaction is a constant and is related to the rate constant by \( t_{1/2} = \frac{0.693}{k} \). In this problem, we need to express the time it takes for the concentration to fall to one-eighth of its initial value, which is equivalent to three half-lives, because \( 2^3 = 8 \).

Connecting Reaction Completion to Half-Lives

To reach one-eighth of the original concentration, the reaction must go through three half-lives. Each half-life reduces the concentration by half. After one half-life, the concentration is \( \frac{1}{2} [A_0] \), after two half-lives \( \frac{1}{4} [A_0] \), and after three half-lives \( \frac{1}{8} [A_0] \).

Calculating Time in Terms of Half-Life

Since each half-life takes time \( t_{1/2} \), three half-lives will take \( 3 imes t_{1/2} \). Thus, the time \( t \) it takes for the concentration to fall to one-eighth its original value is \( 3t_{1/2} \).

Expressing Time in Terms of Rate Constant

Using the relationship between half-life and the rate constant, \( t_{1/2} = \frac{0.693}{k} \), we substitute this into our previous result to get \( t = 3 \times \frac{0.693}{k} = \frac{2.079}{k} \).

Key concepts

Half-Life Calculation

In chemistry, the half-life of a reaction refers to the time required for the concentration of a reactant to decrease to half of its initial value. Understanding half-life is important because it provides insight into how long a reaction will take to proceed to completion.
When dealing with a first-order reaction, the half-life is an easy-to-use concept because it remains constant throughout the reaction process. This means that no matter how much of the reactant is left, the time it takes to reach half of its concentration will always be the same.

• The half-life can be calculated using the formula: \[ t_{1/2} = \frac{0.693}{k} \]where:
  • \( t_{1/2} \) is the half-life of the reaction,
  • \( k \) is the rate constant.

The beauty of the formula lies in its simplicity for first-order reactions. Whether you start with one molar concentration or another, the half-life remains the same, dependent only on the rate constant. Recognizing that reaching one-eighths of the original concentration involves going through three half-lives aids in calculating the total time needed for a reaction.

Rate Constant

The rate constant, denoted by \( k \), is a crucial factor in understanding reaction kinetics. It provides the intrinsic speed at which a reaction occurs, independently of the reactant concentration.
In the context of first-order reactions, the rate constant has units of time inverse, such as \( s^{-1} \) or \( min^{-1} \). This indicates how fast the reaction proceeds over time.
The rate of a first-order reaction can be expressed mathematically as:

• \[ [A] = [A_0] e^{-kt} \]

where:

• \( [A] \) is the concentration of the reactant at time \( t \),
• \( [A_0] \) is the initial concentration,
• \( k \) is the rate constant,
• \( t \) is the time elapsed.

This equation shows the exponential decrease in reactant concentration over time, with \( k \) determining the rate of this decrease. A higher \( k \) means a faster reaction, implying a shorter time to reach half of the initial reactant concentration. Therefore, understanding \( k \) allows you to predict how rapidly a reaction progresses.

Reaction Kinetics

Reaction kinetics provides a framework to study the rates of chemical reactions and factors affecting them. It is an essential part of understanding how reactions occur and are controlled in different environments.
For first-order reactions, kinetics predict that the reaction rate depends solely on the concentration of one reactant, unlike higher-order reactions where multiple reactants might influence the rate. The key equation for first-order kinetics involves using the rate constant \( k \), showing how it directly impacts the rate at which reactants are converted into products.
Within this framework, various concepts such as half-life, rate expressions, and reaction speed interact:

• The rate expression \( -\frac{d[A]}{dt} = k[A] \) represents the change in concentration over time.
• The mathematical relationship between concentration and time is explored using \( [A] = [A_0] e^{-kt} \).
• The concept of half-life simplifies how we understand time progression in reactions.

These elements build a comprehensive picture of how first-order reactions behave over time. By understanding these relationships, scientists and students can predict how long a reaction will take under different conditions and how changes in concentration or rate constant will influence the reaction rate.